Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  relopabVD Structured version   Unicode version

Theorem relopabVD 33434
Description: Virtual deduction proof of relopab 5119. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. relopab 5119 is relopabVD 33434 without virtual deductions and was automatically derived from relopabVD 33434.
1::  |-  (. y  =  v  ->.  y  =  v ).
2:1:  |-  (. y  =  v  ->.  <. x ,. y >.  =  <. x ,. v  >. ).
3::  |-  (. y  =  v ,. x  =  u  ->.  x  =  u ).
4:3:  |-  (. y  =  v ,. x  =  u  ->.  <. x ,. v >.  =  <.  u ,  v >. ).
5:2,4:  |-  (. y  =  v ,. x  =  u  ->.  <. x ,. y >.  =  <.  u ,  v >. ).
6:5:  |-  (. y  =  v ,. x  =  u  ->.  ( z  =  <. x ,. y  >.  ->  z  =  <. u ,  v >. ) ).
7:6:  |-  (. y  =  v  ->.  ( x  =  u  ->  ( z  =  <. x ,.  y >.  ->  z  =  <. u ,  v >. ) ) ).
8:7:  |-  ( y  =  v  ->  ( x  =  u  ->  ( z  =  <. x ,. y  >.  ->  z  =  <. u ,  v >. ) ) )
9:8:  |-  ( E. v y  =  v  ->  E. v ( x  =  u  ->  ( z  =  <. x ,  y >.  ->  z  =  <. u ,  v >. ) ) )
90::  |-  ( v  =  y  <->  y  =  v )
91:90:  |-  ( E. v v  =  y  <->  E. v y  =  v )
92::  |-  E. v v  =  y
10:91,92:  |-  E. v y  =  v
11:9,10:  |-  E. v ( x  =  u  ->  ( z  =  <. x ,. y >.  ->  z  =  <. u ,  v >. ) )
12:11:  |-  ( x  =  u  ->  E. v ( z  =  <. x ,. y >.  ->  z  =  <. u ,  v >. ) )
13::  |-  ( E. v ( z  =  <. x ,. y >.  ->  z  =  <. u  ,  v >. )  ->  ( z  =  <. x ,  y >.  ->  E. v z  =  <. u ,  v >. ) )
14:12,13:  |-  ( x  =  u  ->  ( z  =  <. x ,. y >.  ->  E. v  z  =  <. u ,  v >. ) )
15:14:  |-  ( E. u x  =  u  ->  E. u ( z  =  <. x ,. y  >.  ->  E. v z  =  <. u ,  v >. ) )
150::  |-  ( u  =  x  <->  x  =  u )
151:150:  |-  ( E. u u  =  x  <->  E. u x  =  u )
152::  |-  E. u u  =  x
16:151,152:  |-  E. u x  =  u
17:15,16:  |-  E. u ( z  =  <. x ,. y >.  ->  E. v z  =  <.  u ,  v >. )
18:17:  |-  ( z  =  <. x ,. y >.  ->  E. u E. v z  =  <.  u ,  v >. )
19:18:  |-  ( E. y z  =  <. x ,. y >.  ->  E. y E. u  E. v z  =  <. u ,  v >. )
20::  |-  ( E. y E. u E. v z  =  <. u ,. v >.  ->  E. u E. v z  =  <. u ,  v >. )
21:19,20:  |-  ( E. y z  =  <. x ,. y >.  ->  E. u E. v z  =  <. u ,  v >. )
22:21:  |-  ( E. x E. y z  =  <. x ,. y >.  ->  E. x  E. u E. v z  =  <. u ,  v >. )
23::  |-  ( E. x E. u E. v z  =  <. u ,. v >.  ->  E. u E. v z  =  <. u ,  v >. )
24:22,23:  |-  ( E. x E. y z  =  <. x ,. y >.  ->  E. u  E. v z  =  <. u ,  v >. )
25:24:  |-  { z  |  E. x E. y z  =  <. x ,. y >. }  C_  { z  |  E. u E. v z  =  <. u ,  v >. }
26::  |-  x  e.  _V
27::  |-  y  e.  _V
28:26,27:  |-  ( x  e.  _V  /\  y  e.  _V )
29:28:  |-  ( z  =  <. x ,. y >.  <->  ( z  =  <. x ,. y  >.  /\  ( x  e.  _V  /\  y  e.  _V ) ) )
30:29:  |-  ( E. y z  =  <. x ,. y >.  <->  E. y ( z  =  <. x ,  y >.  /\  ( x  e.  _V  /\  y  e.  _V ) ) )
31:30:  |-  ( E. x E. y z  =  <. x ,. y >.  <->  E. x  E. y ( z  =  <. x ,  y >.  /\  ( x  e.  _V  /\  y  e.  _V ) ) )
32:31:  |-  { z  |  E. x E. y z  =  <. x ,. y >. }  =  {  z  |  E. x E. y ( z  =  <. x ,  y >.  /\  ( x  e.  _V  /\  y  e.  _V ) ) }
320:25,32:  |-  { z  |  E. x E. y ( z  =  <. x ,. y >.  /\  ( x  e.  _V  /\  y  e.  _V ) ) }  C_  { z  |  E. u E. v z  =  <. u ,  v >. }
33::  |-  u  e.  _V
34::  |-  v  e.  _V
35:33,34:  |-  ( u  e.  _V  /\  v  e.  _V )
36:35:  |-  ( z  =  <. u ,. v >.  <->  ( z  =  <. u ,. v  >.  /\  ( u  e.  _V  /\  v  e.  _V ) ) )
37:36:  |-  ( E. v z  =  <. u ,. v >.  <->  E. v ( z  =  <. u ,  v >.  /\  ( u  e.  _V  /\  v  e.  _V ) ) )
38:37:  |-  ( E. u E. v z  =  <. u ,. v >.  <->  E. u  E. v ( z  =  <. u ,  v >.  /\  ( u  e.  _V  /\  v  e.  _V ) ) )
39:38:  |-  { z  |  E. u E. v z  =  <. u ,. v >. }  =  {  z  |  E. u E. v ( z  =  <. u ,  v >.  /\  ( u  e.  _V  /\  v  e.  _V ) ) }
40:320,39:  |-  { z  |  E. x E. y ( z  =  <. x ,. y >.  /\  ( x  e.  _V  /\  y  e.  _V ) ) }  C_  { z  |  E. u E. v ( z  =  <. u ,  v >.  /\  ( u  e.  _V  /\  v  e.  _V ) ) }
41::  |-  { <. x ,. y >.  |  ( x  e.  _V  /\  y  e.  _V  ) }  =  { z  |  E. x E. y ( z  =  <. x ,  y >.  /\  ( x  e.  _V  /\  y  e.  _V ) )  }
42::  |-  { <. u ,. v >.  |  ( u  e.  _V  /\  v  e.  _V  ) }  =  { z  |  E. u E. v ( z  =  <. u ,  v >.  /\  ( u  e.  _V  /\  v  e.  _V ) )  }
43:40,41,42:  |-  { <. x ,. y >.  |  ( x  e.  _V  /\  y  e.  _V  ) }  C_  { <. u ,  v >.  |  ( u  e.  _V  /\  v  e.  _V ) }
44::  |-  { <. u ,. v >.  |  ( u  e.  _V  /\  v  e.  _V  ) }  =  ( _V  X.  _V )
45:43,44:  |-  { <. x ,. y >.  |  ( x  e.  _V  /\  y  e.  _V  ) }  C_  ( _V  X.  _V )
46:28:  |-  ( ph  ->  ( x  e.  _V  /\  y  e.  _V ) )
47:46:  |-  { <. x ,. y >.  |  ph }  C_  { <. x ,. y >.  |  ( x  e.  _V  /\  y  e.  _V ) }
48:45,47:  |-  { <. x ,. y >.  |  ph }  C_  ( _V  X.  _V )
qed:48:  |-  Rel  { <. x ,. y >.  |  ph }
(Contributed by Alan Sare, 9-Jul-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
relopabVD  |-  Rel  { <. x ,  y >.  |  ph }

Proof of Theorem relopabVD
Dummy variables  z 
v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3098 . . . . . 6  |-  x  e. 
_V
2 vex 3098 . . . . . 6  |-  y  e. 
_V
31, 2pm3.2i 455 . . . . 5  |-  ( x  e.  _V  /\  y  e.  _V )
43a1i 11 . . . 4  |-  ( ph  ->  ( x  e.  _V  /\  y  e.  _V )
)
54ssopab2i 4765 . . 3  |-  { <. x ,  y >.  |  ph }  C_  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  _V ) }
63biantru 505 . . . . . . . . . 10  |-  ( z  =  <. x ,  y
>. 
<->  ( z  =  <. x ,  y >.  /\  (
x  e.  _V  /\  y  e.  _V )
) )
76exbii 1654 . . . . . . . . 9  |-  ( E. y  z  =  <. x ,  y >.  <->  E. y
( z  =  <. x ,  y >.  /\  (
x  e.  _V  /\  y  e.  _V )
) )
87exbii 1654 . . . . . . . 8  |-  ( E. x E. y  z  =  <. x ,  y
>. 
<->  E. x E. y
( z  =  <. x ,  y >.  /\  (
x  e.  _V  /\  y  e.  _V )
) )
98abbii 2577 . . . . . . 7  |-  { z  |  E. x E. y  z  =  <. x ,  y >. }  =  { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  ( x  e. 
_V  /\  y  e.  _V ) ) }
10 ax6ev 1736 . . . . . . . . . . . . . . 15  |-  E. u  u  =  x
11 equcom 1780 . . . . . . . . . . . . . . . 16  |-  ( u  =  x  <->  x  =  u )
1211exbii 1654 . . . . . . . . . . . . . . 15  |-  ( E. u  u  =  x  <->  E. u  x  =  u )
1310, 12mpbi 208 . . . . . . . . . . . . . 14  |-  E. u  x  =  u
14 ax6ev 1736 . . . . . . . . . . . . . . . . . . 19  |-  E. v 
v  =  y
15 equcom 1780 . . . . . . . . . . . . . . . . . . . 20  |-  ( v  =  y  <->  y  =  v )
1615exbii 1654 . . . . . . . . . . . . . . . . . . 19  |-  ( E. v  v  =  y  <->  E. v  y  =  v )
1714, 16mpbi 208 . . . . . . . . . . . . . . . . . 18  |-  E. v 
y  =  v
18 idn1 33084 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  (. y  =  v  ->.  y  =  v ).
19 opeq2 4203 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( y  =  v  ->  <. x ,  y >.  =  <. x ,  v >. )
2018, 19e1a 33146 . . . . . . . . . . . . . . . . . . . . . . 23  |-  (. y  =  v  ->.  <. x ,  y
>.  =  <. x ,  v >. ).
21 idn2 33132 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  (. y  =  v ,. x  =  u  ->.  x  =  u ).
22 opeq1 4202 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( x  =  u  ->  <. x ,  v >.  =  <. u ,  v >. )
2321, 22e2 33150 . . . . . . . . . . . . . . . . . . . . . . 23  |-  (. y  =  v ,. x  =  u  ->.  <. x ,  v
>.  =  <. u ,  v >. ).
24 eqeq1 2447 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( <.
x ,  y >.  =  <. x ,  v
>.  ->  ( <. x ,  y >.  =  <. u ,  v >.  <->  <. x ,  v >.  =  <. u ,  v >. )
)
2524biimprd 223 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( <.
x ,  y >.  =  <. x ,  v
>.  ->  ( <. x ,  v >.  =  <. u ,  v >.  ->  <. x ,  y >.  =  <. u ,  v >. )
)
2620, 23, 25e12 33254 . . . . . . . . . . . . . . . . . . . . . 22  |-  (. y  =  v ,. x  =  u  ->.  <. x ,  y
>.  =  <. u ,  v >. ).
27 eqeq2 2458 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( <.
x ,  y >.  =  <. u ,  v
>.  ->  ( z  = 
<. x ,  y >.  <->  z  =  <. u ,  v
>. ) )
2827biimpd 207 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <.
x ,  y >.  =  <. u ,  v
>.  ->  ( z  = 
<. x ,  y >.  ->  z  =  <. u ,  v >. )
)
2926, 28e2 33150 . . . . . . . . . . . . . . . . . . . . 21  |-  (. y  =  v ,. x  =  u  ->.  ( z  = 
<. x ,  y >.  ->  z  =  <. u ,  v >. ) ).
3029in2 33124 . . . . . . . . . . . . . . . . . . . 20  |-  (. y  =  v  ->.  ( x  =  u  ->  ( z  =  <. x ,  y
>.  ->  z  =  <. u ,  v >. )
) ).
3130in1 33081 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  v  ->  (
x  =  u  -> 
( z  =  <. x ,  y >.  ->  z  =  <. u ,  v
>. ) ) )
3231eximi 1643 . . . . . . . . . . . . . . . . . 18  |-  ( E. v  y  =  v  ->  E. v ( x  =  u  ->  (
z  =  <. x ,  y >.  ->  z  =  <. u ,  v
>. ) ) )
3317, 32ax-mp 5 . . . . . . . . . . . . . . . . 17  |-  E. v
( x  =  u  ->  ( z  = 
<. x ,  y >.  ->  z  =  <. u ,  v >. )
)
343319.37aiv 1756 . . . . . . . . . . . . . . . 16  |-  ( x  =  u  ->  E. v
( z  =  <. x ,  y >.  ->  z  =  <. u ,  v
>. ) )
35 19.37v 1755 . . . . . . . . . . . . . . . . 17  |-  ( E. v ( z  = 
<. x ,  y >.  ->  z  =  <. u ,  v >. )  <->  ( z  =  <. x ,  y >.  ->  E. v 
z  =  <. u ,  v >. )
)
3635biimpi 194 . . . . . . . . . . . . . . . 16  |-  ( E. v ( z  = 
<. x ,  y >.  ->  z  =  <. u ,  v >. )  ->  ( z  =  <. x ,  y >.  ->  E. v 
z  =  <. u ,  v >. )
)
3734, 36syl 16 . . . . . . . . . . . . . . 15  |-  ( x  =  u  ->  (
z  =  <. x ,  y >.  ->  E. v 
z  =  <. u ,  v >. )
)
3837eximi 1643 . . . . . . . . . . . . . 14  |-  ( E. u  x  =  u  ->  E. u ( z  =  <. x ,  y
>.  ->  E. v  z  = 
<. u ,  v >.
) )
3913, 38ax-mp 5 . . . . . . . . . . . . 13  |-  E. u
( z  =  <. x ,  y >.  ->  E. v 
z  =  <. u ,  v >. )
403919.37aiv 1756 . . . . . . . . . . . 12  |-  ( z  =  <. x ,  y
>.  ->  E. u E. v 
z  =  <. u ,  v >. )
4140eximi 1643 . . . . . . . . . . 11  |-  ( E. y  z  =  <. x ,  y >.  ->  E. y E. u E. v  z  =  <. u ,  v
>. )
42 19.9v 1741 . . . . . . . . . . . 12  |-  ( E. y E. u E. v  z  =  <. u ,  v >.  <->  E. u E. v  z  =  <. u ,  v >.
)
4342biimpi 194 . . . . . . . . . . 11  |-  ( E. y E. u E. v  z  =  <. u ,  v >.  ->  E. u E. v  z  =  <. u ,  v >.
)
4441, 43syl 16 . . . . . . . . . 10  |-  ( E. y  z  =  <. x ,  y >.  ->  E. u E. v  z  =  <. u ,  v >.
)
4544eximi 1643 . . . . . . . . 9  |-  ( E. x E. y  z  =  <. x ,  y
>.  ->  E. x E. u E. v  z  =  <. u ,  v >.
)
46 19.9v 1741 . . . . . . . . . 10  |-  ( E. x E. u E. v  z  =  <. u ,  v >.  <->  E. u E. v  z  =  <. u ,  v >.
)
4746biimpi 194 . . . . . . . . 9  |-  ( E. x E. u E. v  z  =  <. u ,  v >.  ->  E. u E. v  z  =  <. u ,  v >.
)
4845, 47syl 16 . . . . . . . 8  |-  ( E. x E. y  z  =  <. x ,  y
>.  ->  E. u E. v 
z  =  <. u ,  v >. )
4948ss2abi 3557 . . . . . . 7  |-  { z  |  E. x E. y  z  =  <. x ,  y >. }  C_  { z  |  E. u E. v  z  =  <. u ,  v >. }
509, 49eqsstr3i 3520 . . . . . 6  |-  { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ( x  e.  _V  /\  y  e.  _V )
) }  C_  { z  |  E. u E. v  z  =  <. u ,  v >. }
51 vex 3098 . . . . . . . . . . 11  |-  u  e. 
_V
52 vex 3098 . . . . . . . . . . 11  |-  v  e. 
_V
5351, 52pm3.2i 455 . . . . . . . . . 10  |-  ( u  e.  _V  /\  v  e.  _V )
5453biantru 505 . . . . . . . . 9  |-  ( z  =  <. u ,  v
>. 
<->  ( z  =  <. u ,  v >.  /\  (
u  e.  _V  /\  v  e.  _V )
) )
5554exbii 1654 . . . . . . . 8  |-  ( E. v  z  =  <. u ,  v >.  <->  E. v
( z  =  <. u ,  v >.  /\  (
u  e.  _V  /\  v  e.  _V )
) )
5655exbii 1654 . . . . . . 7  |-  ( E. u E. v  z  =  <. u ,  v
>. 
<->  E. u E. v
( z  =  <. u ,  v >.  /\  (
u  e.  _V  /\  v  e.  _V )
) )
5756abbii 2577 . . . . . 6  |-  { z  |  E. u E. v  z  =  <. u ,  v >. }  =  { z  |  E. u E. v ( z  =  <. u ,  v
>.  /\  ( u  e. 
_V  /\  v  e.  _V ) ) }
5850, 57sseqtri 3521 . . . . 5  |-  { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ( x  e.  _V  /\  y  e.  _V )
) }  C_  { z  |  E. u E. v ( z  = 
<. u ,  v >.  /\  ( u  e.  _V  /\  v  e.  _V )
) }
59 df-opab 4496 . . . . 5  |-  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  _V ) }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  (
x  e.  _V  /\  y  e.  _V )
) }
60 df-opab 4496 . . . . 5  |-  { <. u ,  v >.  |  ( u  e.  _V  /\  v  e.  _V ) }  =  { z  |  E. u E. v
( z  =  <. u ,  v >.  /\  (
u  e.  _V  /\  v  e.  _V )
) }
6158, 59, 603sstr4i 3528 . . . 4  |-  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  _V ) }  C_  { <. u ,  v >.  |  ( u  e.  _V  /\  v  e.  _V ) }
62 df-xp 4995 . . . . 5  |-  ( _V 
X.  _V )  =  { <. u ,  v >.  |  ( u  e. 
_V  /\  v  e.  _V ) }
6362eqcomi 2456 . . . 4  |-  { <. u ,  v >.  |  ( u  e.  _V  /\  v  e.  _V ) }  =  ( _V  X.  _V )
6461, 63sseqtri 3521 . . 3  |-  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  _V ) }  C_  ( _V  X.  _V )
655, 64sstri 3498 . 2  |-  { <. x ,  y >.  |  ph }  C_  ( _V  X.  _V )
66 df-rel 4996 . . 3  |-  ( Rel 
{ <. x ,  y
>.  |  ph }  <->  { <. x ,  y >.  |  ph }  C_  ( _V  X.  _V ) )
6766biimpri 206 . 2  |-  ( {
<. x ,  y >.  |  ph }  C_  ( _V  X.  _V )  ->  Rel  { <. x ,  y
>.  |  ph } )
6865, 67e0a 33302 1  |-  Rel  { <. x ,  y >.  |  ph }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383   E.wex 1599    e. wcel 1804   {cab 2428   _Vcvv 3095    C_ wss 3461   <.cop 4020   {copab 4494    X. cxp 4987   Rel wrel 4994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-opab 4496  df-xp 4995  df-rel 4996  df-vd1 33080  df-vd2 33088
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator